Question

Consider the following statements:
(I) The set \(\{\sin(1/x): x \in (0,1)\}\) is uncountable.
(II) The set \(\{(x,y) \in \mathbb{R}^2 : xy \in \mathbb{Z}\}\) is uncountable.
(III) The set of all \(2 \times 2\) real matrices with rational eigenvalues is uncountable.
Choose the correct answer:

• A. Only (I) is true
• B. Only (I) and (II) are true
• C. Only (II) and (III) are true
• D. All (I), (II) and (III) are true

Hint:

Think about the range of sin over an interval longer than one period, then try explicit uncountable subfamilies for the other two sets.

Answer 1

The Correct Option is D

Step 1: Examine statement (I). Let \(t = 1/x\). As \(x\) ranges over \((0,1)\), \(t\) ranges over \((1,\infty)\). Since \(\sin\) is continuous and periodic with period \(2\pi\), and \((1,\infty)\) contains infinitely many full periods, \(\sin(t)\) takes every value in \([-1,1]\) as \(t\) ranges over \((1,\infty)\). So the given set equals \([-1,1]\) exactly, which is uncountable. Statement (I) is true.

Step 2: Examine statement (II). For each fixed integer \(n\), the level set \(\{(x,y): xy = n\}\) contains uncountably many points (for \(n \neq 0\) it is a hyperbola parametrized by \(x = t, y = n/t\), \(t \neq 0\); for \(n = 0\) it is the union of the two coordinate axes). The set \(\{(x,y): xy \in \mathbb{Z}\}\) is the union of these level sets over all \(n \in \mathbb{Z}\), which is a countable union of uncountable sets, hence uncountable. Statement (II) is true.

Step 3: Examine statement (III). Take the family of matrices \(A_b = \begin{pmatrix} 0 & b \\ 0 & 0 \end{pmatrix}\) for \(b \in \mathbb{R}\). The characteristic polynomial is \(\lambda^2 = 0\), so both eigenvalues equal \(0\), which is rational, for every value of \(b\). Since \(b\) ranges over all of \(\mathbb{R}\), this gives uncountably many distinct \(2 \times 2\) real matrices with rational eigenvalues. Statement (III) is true.

Step 4: All three statements (I), (II) and (III) are true.

\[\boxed{\text{All (I), (II) and (III) are true}}\]